SCIENTIA SINICA Informationis, Volume 50 , Issue 10 : 1501(2020) https://doi.org/10.1360/N112018-00260

## A warning propagation algorithm to solve the double-objective minimum spanning tree

• AcceptedAug 13, 2019
• PublishedOct 12, 2020
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### Funded by

2018宁夏回族自治区重点研发计划项目(2018BEE03019)

### References

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• Figure 1

An instance of factor graph

• Figure 2

A model transformation instance of RBM

• Figure 3

Schematic diagram of transformation principle

• Table 1   Parameters of 6 groups of random undirected graphs with different numbers of nodes
 $n$ $m$ $\alpha$ $d$ 100 4950 49.5 3.475 150 11175 74.5 3.475 200 19900 99.5 3.475 300 44850 149.5 3.475 350 61075 174.5 3.475 400 79800 199.5 3.475
•

Algorithm 1 Factor graph transformation algorithm

Require:Random bigraph instance $G$;

Output:Factor graph $F$;

A random bigraph instance is selected;

Select the initial point randomly $v\in~V$ as the starting point of transformation;

while $v~\le~V$ do

if $v$ is not the isolated point then

The bigraph edge $e\in~E$ is mapped to a clause node of the factor graph;

Two sets of weights on the edge of a bigraph are mapped to two nodes of a factor graph;

else

Draw auxiliary edges to connect outliers;

$v=v+1$;

end if

end while

Add a binary tag bit to each edge and map it to the third argument node;

•

Algorithm 2 Warning propagation algorithm for solving double objective minimum spanning tree

Require:Factor graph $F$; Maximum number of iterations $t_{\rm~max}$; a requested precision $\varepsilon$;

Output:The number of nodes in the minimum spanning tree $\eta~_{a\rightarrow~i}^{*}$;

Get the factor graph by Algorithm 1;

A node is randomly selected as the initial iteration node $n_{i}\in~V$;

At time $t\leftarrow~0$, for every edge of the factor graph init the warning messages $W_{a\rightarrow~i}\in\left~\{~0,~1~\right~\}$;

Let us do a random permutation of the edges in $G$;

Using 2 to update the warning information until the algorithm converges;

while $t~\le~t_{\rm~max}$ do

Sweep the set of edges in the factor graph and update the warning information;

if $\left~|\eta~_{a\rightarrow~i}(t)-\eta~_{a\rightarrow~i}(t-1)~~\right~|<~\varepsilon~$ then

Takes the convergent nodes to form the nodes of the double-objective minimum spanning tree;

else

Continue;

end if

Break;

end while

The sum of the weights of two variable nodes representing weights in the factor graph is calculated respectively;

Get part of the assignment;

• Table 2   Comparison of the running time of MACS algorithm and WP algorithm (s)
 The number of node Enumeration MACS WP 20 3360 552. 6 20. 423 30 8700 681 30. 697 40 20580 847. 2 40. 537 50 45900 1095. 6 51. 018

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